This article shows how Pythagorean triples can be generated naturally from a class of infinite series whose sums are zero, making surprising and interesting connections between different areas mathematics. This material can be suitable for use by teachers and students at various levels, but the simplest forms of the ideas may be best understood at the beginning of a post-calculus course on complex numbers. More detailed and advanced aspects of this problem can be considered for appropriate students. The paper begins by determining the sum of two trigonometric series.